A product-sum maximum on two circles

Proved

WorkbookCorrected.plus_72159

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

corrected-formalizationlean-workbooksource-checked

Let $a^{2}+b^{2}=x^{2}+y^{2}=2$ then show that maximum value of $S=(1-a)(1-b)+(1-x)(1-y)$ is $8$

Source: InternLM Lean-Workbook, record lean_workbook_plus_72159 (Apache-2.0). Natural-language proposition preserved; the source bound is completed with an exact attainment witness. Proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookCorrected.plus_72159 : (∀ (a b x y : ℝ) (ha : a^2 + b^2 = 2) (hb : x^2 + y^2 = 2), (1 - a) * (1 - b) + (1 - x) * (1 - y) ≤ 8) ∧ (∃ a b x y : ℝ, (a^2 + b^2 = 2) ∧ (x^2 + y^2 = 2) ∧ ( (1 - a) * (1 - b) + (1 - x) * (1 - y)  =  8   )) := by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_72159; Apache-2.0

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